Degenerating curves and surfaces: first results

Let S → A be a smooth family of surfaces whose general fibre is a smooth surface of P3 and whose special fibre has two smooth components, intersecting transversally along a smooth curve R. We consider the Universal Severi-Enriques variety V on S → A. The general fibre of V is the variety of curves on St in the linear system |OSt (n)| with k cusps and δ nodes as singularities. Our problem is to find all irreducible components of the special fibre of V . In this paper, we consider only the cases (k, δ) = (0, 1) and (k, δ) = (1, 0). In particular, we determine all singular curves on the special fibre of S which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of S.